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The importance Markov chain

Author

Listed:
  • Andral, Charly
  • Douc, Randal
  • Marival, Hugo
  • Robert, Christian P.

Abstract

The Importance Markov chain is a novel algorithm bridging the gap between rejection sampling and importance sampling, moving from one to the other through a tuning parameter. Based on a modified sample of an instrumental Markov chain targeting an instrumental distribution (typically via a MCMC kernel), the Importance Markov chain produces an extended Markov chain where the marginal distribution of the first component converges to the target distribution. For example, when targeting a multimodal distribution, the instrumental distribution can be chosen as a tempered version of the target which allows the algorithm to explore its modes more efficiently. We obtain a Law of Large Numbers and a Central Limit Theorem as well as geometric ergodicity for this extended kernel under mild assumptions on the instrumental kernel. Computationally, the algorithm is easy to implement and preexisting librairies can be used to sample from the instrumental distribution.

Suggested Citation

  • Andral, Charly & Douc, Randal & Marival, Hugo & Robert, Christian P., 2024. "The importance Markov chain," Stochastic Processes and their Applications, Elsevier, vol. 171(C).
    • Handle: RePEc:eee:spapps:v:171:y:2024:i:c:s030441492400022x
    DOI: 10.1016/j.spa.2024.104316
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    References listed on IDEAS

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    1. James Berger & Elías Moreno & Luis Pericchi & M. Bayarri & José Bernardo & Juan Cano & Julián Horra & Jacinto Martín & David Ríos-Insúa & Bruno Betrò & A. Dasgupta & Paul Gustafson & Larry Wasserman &, 1994. "An overview of robust Bayesian analysis," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 3(1), pages 5-124, June.
    2. Raices Cruz, Ivette & Lindström, Johan & Troffaes, Matthias C.M. & Sahlin, Ullrika, 2022. "Iterative importance sampling with Markov chain Monte Carlo sampling in robust Bayesian analysis," Computational Statistics & Data Analysis, Elsevier, vol. 176(C).
    3. Liu J. S & Liang F. & Wong W.H., 2001. "A Theory for Dynamic Weighting in Monte Carlo Computation," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 561-573, June.
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